lectures on commutative algebra - mcm › ~zheng › commalg.pdflet rbe a noetherian domain. the...

Lectures on Commutative Algebra

Weizhe Zheng

Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory ofMathematicsAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijing 100190, China

University of the Chinese Academy of Sciences, Beijing 100049, China

Email: [email protected]

Contents

Convention v

7 UFDs 1

8 Primary decomposition 3

9 DVRs and Dedekind domains 5

10 Completions 11

11 Dimension theory 17

Summary of properties of rings 23

iii

iv CONTENTS

Convention

Rings are assumed to be commutative.

v

vi CONVENTION

Chapter 7

UFDs

Theorem 7.1 (Fundamental theorem of arithmetic, cf. Euclid’s Elements 9.14).Every nonzero integer n ∈ Z can be factorized uniquely (up to permutation of factors)as a product of primes n = ±p1 · · · pm.

Definition 7.2. Let R be a domain.(1) x ∈ R is irreducible if x 6= 0, x 6∈ R×, and x = yz implies y ∈ R× or z×.(2) x, y ∈ R are associates if there exists u ∈ R× such that x = uy.(3) R is a unique factorization domain (UFD, or factorial ring) if every x ∈ R

satisfying x 6= 0 and x 6∈ R admits a factorization x = a1 · · · am with aiirreducible, and if x = b1 · · · bn with bj irreducible, thenm = n and there existsa bijection σ : {1, . . . , n} → {1, . . . , n} such that ai and bσ(i) are associates forevery i).

We say that x ∈ R is prime if xR is a prime ideal. Prime elements are irreducible.The converse holds in a UFD.

Theorem 7.3. Let R be a domain. Then R is a UFD if and only if the followingconditions are satisfied:(1) The ascending chain condition for principle ideals of R.(2) Irreducible elements in R are prime.

Lemma 7.4. Assume that the ascending chain condition holds for principle idealsof a ring R. Then every x ∈ R satisfying x 6= 0 and x 6∈ R admits a factorizationx = a1 · · · am with ai irreducible.

Corollary 7.5. A PID is a UFD.

Example 7.6. (1) Z, Z[√−1] are PIDs, hence UFDs.

(2) Z[√−5] is not a UFD. Indeed, 2 · 3 = 6 = (1 +

√−5)(1−

√−5).

(3) If R is a UFD, R[X] is a UFD. Note however that neither Z[X] nor k[X, Y ](where k is a field) is a PID.

(4) If R is a nonzero UFD, R[X1, X2, . . . ] is a UFD, but not a Noetherian ring.(5) In R = ⋃∞

n=1 k[X1/n], factorization does not exist in general. In particular, Rdoes not satisfy the ascending chain condition for principal ideals.

1

2 CHAPTER 7. UFDS

Chapter 8

Primary decomposition

We largely followed [AM, Chapters 4 and 7]. Most other sources define primesassociated to an ideal I of a ring R differently, as prime ideals of the form (I : x).The two definitions coincide when R is Noetherian.

3

4 CHAPTER 8. PRIMARY DECOMPOSITION

Chapter 9

DVRs and Dedekind domains

We studied Artinian rings, which are Noetherian rings of dimension 0. In thischapter, we study the next simplest case, Noetherian domains of dimension 1. Westart by applying primary decomposition to such domains.

Proposition 9.1. Let R be a Noetherian domain of dimension 1. Every nonzeroideal I ⊆ R can be uniquely written as a product of primary ideals whose radicalsare distinct.

We would like to further decompose primary ideals into prime powers. We firstlook at the local case.

Discrete valuation ringsRecall that the value group of a valuation ring R is K×/R×, where K = Frac(R).

Definition 9.2. A discrete valuation ring (DVR) is a valuation ring whose valuegroup is isomorphic to Z.

Proposition 9.3. Let R be a valuation ring. The following are equivalent:(1) R is a DVR.(2) R is a PID.(3) R is Noetherian.

For (2) ⇒ (1), we use the following.

Lemma 9.4. Let R be a local ring of dimension > 0. Assume that the maximalideal m of R is principal and ⋂∞n=1 m

n = 0. Then R is a DVR. The assumption⋂∞n=1 m

n = 0 holds if R is a Noetherian domain.

We will see later that the assumption ⋂∞n=1 mn = 0 holds for R Noetherian.

A generator of the maximal ideal of a DVR is called a uniformizer.Example 9.5. (1) Let K = Q and let p be a rational prime. The p-adic valuation

vp : K× → Z is defined by vp(pa uv ) = a, where a, u, v ∈ Z, (u, p) = (v, p) = 1.Every valuation of K× is equivalent to vp (Ostrowski’s theorem).

5

6 CHAPTER 9. DVRS AND DEDEKIND DOMAINS

(2) Let k be a field and let K = k(X). Similarly to (1), for every irreduci-ble polynomial f ∈ k[X], the f -adic valuation vf : K×× → Z is defined byvf (fa uv ) = 1, where a ∈ Z, u, v ∈ k[X], (u, f) = (v, f) = 1. Every valuation ofK× is equivalent to vf or v1/X .

(3) Let k be a field and let K = ⋃n k(X1/n). We have a non-discrete rank 1

valuation v : K× → Q whose restriction to k(X1/n) is 1nvX1/n with vX1/N defined

in (2). The valuation ideal m = (X1/n)n≥1 is not finitely generated.(4) Let F be a field and let vF : F× → Γ. Let K = F (X) (or F ((X))). Then

K× → Z × Γ carrying f = ∑n≥N anX

n with aN 6= 0 to (N, v(aN)) is avaluation, of rank > 1 for vF nontrivial. Here Z × Γ is equipped with thelexicographical order.

(5) Consider the particular case of (4) where F = k(Y ) and vF = vY . Let m bethe valuation ideal. Then m = Y R is principal, but ⋂nmn = (X/Y n)n≥1 isnot finitely generated.

Definition 9.6. We say that a ring R is normal if for every prime p, Rp is anintegrally closed domain.

A normal domain is synonymous to an integrally closed domain.

Proposition 9.7. Let R be a Noetherian local domain that is not a field. Let m bethe maximal ideal and let k = m/m2. The following conditions are equivalent:(1) R is a DVR.(2) R is normal of dimension one.(3) m is principal.(4) dimk(m/m2) = 1.(5) Every nonzero ideal of R is a power of m.

For (2)⇒(3), we use the following.

Lemma 9.8. Let R be a normal local domain and assume that the maximal idealm is finitely generated and there exists a, b ∈ R such that m = (aR : b). Then m isprincipal.

The proposition holds in fact for R a Noetherian local ring of dimension > 0.The proof uses the Krull intersection theorem (Corollary 10.28).

Dedekind domainsTheorem 9.9. Let R be a Noetherian domain of dimension one. The following areequivalent:(1) R is normal.(2) Every primary ideal of R is a power of a prime ideals.(3) Every local ring Rp (p 6= 0) is a DVR.

Definition 9.10. A Dedekind domain is a Noetherian domain of dimension onesatisfying the above equivalent conditions.

Corollary 9.11. Every nonzero ideal of a Dedekind domain has a unique factori-zation as a product of prime ideals.

7

Remark 9.12. The following converse of Corollary 9.11 holds: An integral domainof which every nonzero ideal is a product of prime ideals is a Dedekind domain[M2, Theorem 11.6].Example 9.13. A PID is a Dedekind domain.

More examples are given by taking integral closure.

Proposition 9.14. Let A be a normal domain. Let L be a finite separable extensionof K = Frac(A) and let B be the integral closure of A in L. Then B is contained ina finitely generated A-submodule of L.

Corollary 9.15. Let A be Dedekind domain. Let L be a finite separable extensionof K = Frac(A). Then the integral closure B of A in L is a Dedekind domain.

Remark 9.16. More generally, if A is a Noetherian domain of dimension 1 (notnecessarily normal) and if L is a finite (not necessarily separable) extension of K =Frac(A), then the normalization B of A in L is a Dedekind domain (even when Bis not finite over A). This follows from the Krull-Akizuki Theorem [M2, Theorem11.7].Example 9.17. Let K be a number field (namely, a finite extension of Q). Then thering of integers OK (namely, the integral closure of Z in K) is a Dedekind domainby Corollary 9.15.Example 9.18. In particular, for K = Q(

√−5), R = Z[

√−5] is a Dedekind domain.

The prime ideal p = (2, 1 +√−5) is not principal. Indeed, since p2 = 2R, if

p = αR for α = a + b√−5, a, b ∈ Z, then (NK/Qα)2 = NK/Q2 = 4, so that

a2 + 5b2 = NK/Qα = 2, which is impossible.We have 3R = qq′, (1+

√−5)R = pq, (1−

√−5)R = pq′, where q = (3, 1+

√−5),

q′ = (3, 1−√−5).

The maximal ideal of the DVR Rp is (1 +√−5)Rp.

Example 9.19. Let R = Z[√

5]. This is a Noetherian domain of dimension one, butnot normal. The integral closure of R in Q(

√5) is Z[1+

√5

2 ]. Consider the prime idealp = (2, 1 +

√5) of R. We have p2 = 2p ⊆ 2R ⊆ p. The ideal 2R is p-primary, but

not a power of p.

Definition 9.20. We say that a domain A is Japanese (or N-2) if for every finiteextension L of K = Frac(A), the integral closure of A in L if finite over A. Wesay that a Noetherian ring R is Nagata (or universally Japanese) if every finitelygenerated integral R-algebra is Japanese.

Grothendieck uses “Japanese” and “universally Japanese” [G, Sections 0.23,IV.7.6, IV.7.7], while Matsumura uses “N-2” and “Nagata” [M1, Chapter 12].Example 9.21. (1) Every normal domain of perfect fraction field is Japanese by

Proposition 9.14.(2) Every field is Nagata. Every Dedekind domain of fraction field of characteristic

0 is Nagata.(3) Nagata constructed a DVR that is not Nagata (not Japanese?).


Invertible modulesThe nonzero ideals of Dedekind domain is a free commutative monoid with maximalideals forming a basis. We now consider the associated free abelian group. Thereare a number of generalizations to commutative rings.

Proposition 9.22. Let R be a ring M be a finitely presented module. The followingare equivalent:(1) For every prime ideal p of R, the Rp-module Mp is isomorphic to Rp.(2) For every maximal ideal m of R, the Rm-module Mm is isomorphic to Rm.(3) The evaluation map u : M∗ ⊗R M → R is an isomorphism, where M∗ =

HomR(M,R).(4) There exists an R-module N such that M ⊗R N is isomorphic to R.

Definition 9.23. A finitely generated R-module M is said to be invertible if itsatisfies the above conditions. The Picard group Pic(R) of a ring R is the abeliangroup of isomorphism classes of invertible R-modules M with group law given bytensor product. The class of M is denoted cl(M).

The identity element of Pic(R) is cl(R) and cl(M)−1 = cl(M∗).Remark 9.24. The proposition holds more generally for M finitely generated. Theequivalent conditions then imply that M is projective and finitely presented. Inver-tible R-modules are also called projective R-modules of rank 1. See [B2, II.5].Remark 9.25. For a local ring R, Pic(R) = 0.

Let R be a domain and let K = Frac(R).

Lemma 9.26. Every invertible R-module M is an R-submodule of K.

Definition 9.27. An R-submodule of K is called a fractional ideal of R if thereexists x ∈ R, x 6= 0 such that xI ⊆ R.

Example 9.28. (1) Every ideal of R is a fractional ideal of R.(2) For every x ∈ K, xR is a fractional ideal of R. Such fractional ideals are said

to be principal. A principal fractional ideal is free of rank ≤ 1. Conversely,any free fractional ideal is principal.

Remark 9.29. Every finitely generated R-submodule of K is a fractional ideal. Con-versely, if R is Noetherian, then every fractional ideal if finitely generated.

R-submodules of K form a commutative monoid, with identity element R andIJ = {∑n

i=1 aibi | ai ∈ I, bi ∈ J}. We write I−1 = {x ∈ K | xI ⊆ R}.

Proposition 9.30. Let I be an R-submodule of K. The following are equivalent:(1) I is finitely generated and for every prime ideal p, Ip ' Rp.(2) I is finitely generated and for every maximal ideal m, Im ' Rm.(1′) I is finitely generated and for every prime ideal p, Ip(Ip)−1 ' Rp.(2′) I is finitely generated and for every maximal ideal m, Im(Im)−1 ' Rm.(3) II−1 = R.(4) There exists an R-submodule J of K such that IJ = R.

9

Definition 9.31. An invertible (fractional) ideal is a fractional ideal satisfying theabove conditions. We let CaDiv(R) (for Cartier divisors) denote the abelian groupof invertible ideals.

Proposition 9.32. We have a exact sequence 1 → R× → K×·R−→ CaDiv(R) cl−→

Pic(R)→ 1.

Remark 9.33. For K a number field, O×K is a finitely generated abelian group andPic(O×K) is a finite abelian group, called the class group of K.

Proposition 9.34. Let R be a UFD. Then Pic(R) = 1.

Lemma 9.35. Let R be a Noetherian domain and let p be an invertible prime ideal.Then Rp is a DVR.

Theorem 9.36. Let R be a domain that is not a field. Then the following areequivalent:(1) R is a Dedekind domain.(2) Every nonzero ideal of R is invertible.(3) Every nonzero fractional ideal of R is invertible.

Corollary 9.37. A Dedekind UFD is a PID.

Proposition 9.38. Let R be a Dedekind domain. Then CaDiv(R) is a free abeliangroup with maximal ideals forming a basis.

Definition 9.39. The height of a prime ideal p of a ring R is the supremum of thelength n of chains p0 ( p1 ( · · · ( pn = p of prime ideals.

Remark 9.40. We have ht(p) = dim(Rp), dim(R) ≤ ht(p)+dim(R/p), and dim(R) =supp ht(p) = supp dim(R/p).Remark 9.41. (1) For any ring R, one can define an abelian group CaDiv(R) and

there is an exact sequence 1→ R× → K× → CaDiv(R)→ Pic(R).(2) For any ring R, the group Div(R) of Weil divisors is defined to the free abelian

group generated by the primes ideals of p of height 1. For R a Noetheriandomain, there is a homomorphism CaDiv(R)→ Div(R), which is injective forR normal and an isomorphism for R locally factorial (namely, Rp is a UFD forall p).

We end this chapter with a criterion of normality.

Proposition 9.42. Let R be a Noetherian domain. The following conditions areequivalent:(1) R is normal.(2) For every prime ideal p associated to a nonzero principal ideal of R, Rp is a

DVR.(3) R = ⋂

ht(p)=1Rp and for each p of height 1, Rp is a DVR.

For (2)⇒(3), we use the following.

Lemma 9.43. Let R be a Noetherian domain. Then R = ⋂pRp, where p runs

through prime ideals associated to nonzero principal ideals of R.

Chapter 10

Completions

Topologies and completionsDefinition 10.1. A topological group is a group G equipped with a topology suchthat the maps

G×G→ G (x, y) 7→ xy (multiplication)G→ G x 7→ x−1 (inversion)

are continuous. A topological ring is a ring R equipped with a topology such thatthe maps

R×R→ R (x, y) 7→ x+ y

R×R→ R (x, y) 7→ xy

are continuous. A topological R-module is an R-moduleM equipped with a topologysuch that the maps

M ×M →M (x, y) 7→ x+ y

R×M →M (r, x) 7→ rx

are continuous.Let G be a topological abelian group. In the sequel we write the group law

additively. For any a ∈ G, the translation map Ta : G → G, g 7→ g + a is ahomeomorphism.Lemma 10.2. Let H be the intersection of neighborhoods of 0 in G. Then H is asubgroup of G, closure of {0}. Moreover, the following conditions are equivalent:(1) G is Hausdorff.(2) Every point of G is closed.(3) H = 0.

Proof. That H is a subgroup of G follows from the continuity of group operations.For x ∈ G, x ∈ H if and only if 0 ∈ x − U for all neighborhoods U of 0, which isequivalent to x ∈ {0}. Then (3) is equivalent to 0 being a closed point of G. Thus(1)⇒(2)⇒(3). Conversely, if 0 is a closed point of G, then the diagonal ∆ ⊆ G×Gis a closed subset, namely G is Hausdorff. Indeed, ∆ = d−1(0), where d : G×G→ Gis the continuous map defined by d(x, y) = x− y.

11

12 CHAPTER 10. COMPLETIONS

To define the completion of a topological abelian group in full generality, we needthe following generalization of sequences. We will soon restrict to cases where thetopology is first-countable, for which sequences suffices.

Definition 10.3. A directed set is a set I equipped with a preorder ≤ (a reflexiveand transitive binary relation: i ≤ i; i ≤ j and j ≤ k implies i ≤ k) such that foreach pair i, j ∈ I, there exists k ∈ I such that i ≤ k and j ≤ k.

A net in a set X is a collection (xi)i∈I of elements of X, where I is a directedset. Given a subset U ⊆ X, we say that (xi)i∈I eventually belongs to U if thereexists i0 ∈ I such that xi ∈ U for all i ≥ i0. A net (xi)i∈I in a topological spaces Xconverges to x ∈ X if for it eventually belongs to every neighborhood U of x in X.

Limits of nets in X are unique (whenever they exists) if and only if every pointof X is closed.

Definition 10.4. Let G be a topological abelian group. A net (xi)i∈I in G is calleda Cauchy net if for every neighborhood U of 0 ∈ G, there exists i0 ∈ I such that forall i, j ≥ i0, xi − xj ∈ U (in other words, the net (xi − xj)(i,j)∈I×I converges to 0).We say that G is complete if every Cauchy net converges to a unique point of G.

As usual, convergent nets are Cauchy nets. Our definition of complete includesHausdorff (unlike in Bourbaki).

Proposition 10.5. Let G be a topological abelian group. There exists a topologicalabelian group G and a continuous homomorphism φ : G → G such that for everycontinuous homomorphism f : G→ H of topological abelian groups with H complete,there exists a unique continuous homomorphism g : G→ H such that f = gφ.

G is called the completion of G.Remark 10.6. (1) Similar results hold for topological rings and modules, but not

for topological groups [B1, Exercice X.3.16].(2) By the universal property, G is unique up to unique isomorphism (of topolo-

gical groups). The image of φ is dense. Moreover, the assignment G 7→ G isfunctorial.

(3) By the proof, ker(φ) is the intersection of the neighborhoods of 0 ∈ G. ThusG is Hausdorff if and only if φ is injective.

(4) By the proof, if H is a subgroup of G equipped with the subspace topology,then H can be identified with a topological subgroup of G. The subgroupH < G is closed (since H is complete and G is Hausdorff) and is the closureof the image of H → G.

The following is an immediate consequence of the universal property.

Corollary 10.7. φG : G→ ˆG is an isomorphism (of topological groups).

Assume that 0 ∈ G admits a fundamental system of neighborhoods consisting ofsubgroups (Gλ)λ∈Λ of G (indexed by inverse inclusion). Note that Gλ is open. Theprojection G → G/Gλ induces G → G/Gλ, with G/Gλ equipped with the discretetopology.

13

Proposition 10.8. The map G → limλG/Gλ is an isomorphism of topologicalgroups.

Here limλG/Gλ ⊆∏λG/Gλ is equipped with the subspace topology.

Proposition 10.9. Let 0 → G′ → Gπ−→ G′′ → 0 be an exact sequence of abelian

groups. Equip G′ with the subspace topology and G′′ with the quotient topology. Thenwe have a an exact sequence of groups 0→ G′ → G

π−→ G′′. Moreover, π is surjectiveif Λ = Z≥0.

Assume Λ = Z≥0 in the sequel.

Corollary 10.10. We have G/Gn ' G/Gn.

Let R be a ring and letM be an R-module. The I-adic topology on R is given byR ⊇ I ⊇ I2 ⊇ . . . , and the I-adic topology onM is given byM ⊇ IM ⊇ I2M ⊇ . . . .Example 10.11. (1) For R = Z and I = pZ, R = Zp is the ring of p-adic integers.(2) For R = R0[X1, . . . , Xn] and I = (X1, . . . , Xn), R = R0[[X1, . . . , Xn]] is the

ring of formal power series.(3) For R = Zp[X1, . . . , Xn] and I = (p), R = Zp〈X1, . . . , Xn〉 is the ring of

convergent power series ∑(i1,...,in)∈Zn≥0ai1,...,inX

i11 · · ·X in

n satisfying ai1,...,in → 0for i1 + · · ·+ in →∞.

(4) Let k be field of characteristic 6= 2 and R = k[x, y]/(y2 − (1 + x)), I = (x).Then R ' k[[x]] ⊕ k[[x]], carrying y to (

√1 + x,−

√1 + x), where

√1 + x =

1+ x2−

x2

8 + · · · ∈ k[[x]]. An important generalization of this is Hensel’s Lemma(exercise).

(5) Let k be as above and R = k[x, y]/(y2 − x2(1 + x)), I = (x). Then R 'k[[x]][y]/(y − x

√1 + x)(y + x

√1 + x).

Proposition 10.12. Let R be a ring and let I ⊆ R be an ideal such that R isI-adically complete. Then I ⊆ rad(R).

FiltrationsLetM be an R-module and letM = M0 ⊇M1 ⊇M2 ⊇ . . . be a decreasing filtrationby R-submodules.

Definition 10.13. Let I ⊆ R be a ideal. We say that the filtration (Mi) is anI-filtration if IMn ⊆ Mn+1 for all n ≥ 0. We say that the filtration (Mi) is a stableI-filtration if moreover there exists N such that IMn = Mn+1 for all n ≥ N .

Lemma 10.14. Let (Mn) and (M ′n) be stable I-filtrations of M . Then they have

bounded difference: There exists an integer N ≥ 0 such that Mn+N ⊆ M ′n and

M ′n+N ⊆ Mn for all n ≥ 0. Hence stable I-filtrations determine the same topology

on M as the I-adic topology.

Proof. We may assume M ′n = InM . Then InM = InM0 ⊆ Mn for all n ≥ 0. If

IMn = Mn+1 for all n ≥ N , then Mn+N = InMN ⊆ InM for all n ≥ N .


Proposition 10.15 (Artin-Rees lemma). Let R be a Noetherian ring, I ⊆ R anideal, M a finitely generated R-module, (Mn) a stable I-filtration of M , M ′ ⊆ Man R-submodule. Then (M ′ ∩Mn) is a stable I-filtration of M ′. In particular, thereexists an integer N ≥ 0 such that (IN+nM) ∩M ′ = In((INM) ∩M ′) for all n ≥ 0.

Corollary 10.16. The I-adic topology on M ′ coincides with the subspace topologyinduced from the I-adic topology of M .

We will prove the Artin-Rees lemma after some constructions.

Definition 10.17. A graded ring is a ringR together with an isomorphism of abeliangroups R ' ⊕∞n=0 Rn such that RmRn ⊆ Rm+n for all m,n ≥ 0. A graded R-moduleis an R-module together with an isomorphism of abelian groupsM '⊕∞n=0Mn suchthat RmMn ⊆Mm+n.

It follows that R0 is a ring and each Rn and each Mn are R0-modules.An element x ∈ M is said to be homogeneous if x ∈ Mn for some n. For

x = ∑n xn with xn ∈Mn, the xn’s are called the homogeneous components of x.

Definition 10.18. Let R be a ring and let I ⊆ R be an ideal. The blowup algebrais the graded ring (in fact a graded R-algebra) BIR = ⊕∞

n=0 In. For an R-module

and an I-filtration F = (Mn), we define the graded BIR-module BFM = ⊕∞n=0Mn.

Proposition 10.19. (1) Assume Mn is finitely generated R-module for all n ≥ 0.Then BFM is a finitely generated BIR-module if and only if F is I-stable.

(2) Assume that R is a Noetherian ring. Then BIR is a Noetherian ring.

The Artin-Rees lemma has many consequences.

Proposition 10.20. Let R be a Noetherian ring and let 0→M ′ →M →M ′′ → 0be an exact sequence of finitely generated R-modules. For any ideal I ⊆ R, takingI-adic completion gives an exact sequence 0→ M ′ → M → M ′′ → 0.

Proposition 10.21. Let R be a ring, I ⊆ R an ideal, M a finitely generatedR-module. The homomorphism φ : M → M induces a surjection ψ : R ⊗R M →M , where ˆ denotes the I-adic completion. In particular, M = Rφ(M). For MNoetherian, ψ is an isomorphism.

Corollary 10.22. Let R be a ring and I ⊆ R a finitely generated ideal. ThenIn = In. Moreover, for any finitely generated R-module M , M has the I-adictopology as an R-module and the I-adic topology as an R-module.

Corollary 10.23. Let R be a ring and let m ⊆ R be a finitely generated maximalideal. Then the m-adic completion R is a local ring of maximal ideal m. Moreover,R→ R factorizes through Rm → R that identifies R as the mRm-adic completion ofRm.

Corollary 10.24. Let R be a Noetherian ring and I ⊆ R an ideal. Then the I-adiccompletion R is a flat R-algebra.

15

Theorem 10.25 (Krull). Let R be a Noetherian ring, I ⊆ R an ideal, M a finitelygenerated R-module. Then Ker(M → M) = ⋂∞

n=0 InM consists of those x ∈ M

killed by some r ∈ 1 + I.

Corollary 10.26. Let R be a Noetherian domain and let I ( R be a proper ideal.Then ⋂∞n=0 I

n = 0.

Corollary 10.27. Let R be a Noetherian ring, I ⊆ rad(R) and ideal of R, Ma finitely generated R-module. Then ⋂

n=0 InM = 0. In other words, the I-adic

topology on M is Hausdorff. Moreover, every R-submodule M ′ of M is closed.

Corollary 10.28. Let R be a Noetherian local ring and let m be the maximal idealof R. Then ⋂∞n=0 m

n = 0.

This allows to extend Proposition 9.7 to Noetherian local rings.

Proposition 10.29. Let R be a Noetherian ring and I ⊆ R an ideal. The followingconditions are equivalent:(1) I ⊆ rad(R).(2) Every ideal of R is closed for the I-adic topology.(3) The I-adic completion R of R is a faithfully flat R-algebra.

We conclude this chapter with the following.

Theorem 10.30. Let R be a Noetherian ring and I ⊆ R an ideal. The I-adiccompletion R of R is a Noetherian ring.

Corollary 10.31. Let R be a Noetherian ring. Then the ring R[[X1, . . . , Xn]] offormal power series is Noetherian.

Associated graded ringsDefinition 10.32. Let R be a ring, I ⊆ R an ideal. The associated graded ringgrIR = ⊕∞

n=0 In/In+1. For a graded R-module M with a I-filtration F = (Mn), the

associated graded grIR-module grFM = ⊕∞n=0Mn/Mn+1.

Remark 10.33. The associated graded ring is related to the blowup algebra by grIR 'BIR⊗R R/I. Similarly, grFM ' BFM ⊗R R/I.

Proposition 10.34. (1) If every Mn is a finitely generated R-module and F isI-stable, then grFM is a finitely generated grIR-module.

(2) If R is a Noetherian ring, then grIR is a Noetherian ring and grIR ' grIR.

The proof of Theorem 10.30 relies on a partial converse of the implication RNoetherian ⇒ grIR Noetherian.

Lemma 10.35. Let φ : A→ B be a homomorphism of filtered abelian groups.(1) If gr(φ) is injective, then φ is injective.(2) If gr(φ) is surjective, then φ is surjective.


Proposition 10.36. Let R be a ring, I ⊆ R an ideal such that R is I-adicallycomplete, M an R-module, F = (Mn) an I-filtration of M such that ⋂nMn = 0. IfgrFM is a finitely generated grIR-module, then M is a finitely generated R-module.

Corollary 10.37. If grFM is a Noetherian grIR-module, then M is a NoetherianR-module.

Chapter 11

Dimension theory

Hilbert functionsProposition 11.1. Let R = ⊕∞

n=0 Rn be a graded ring. Then R is Noetherian ifand only if R0 is Noetherian and R is a finitely generated R0-algebra.

Corollary 11.2. Let R = ⊕∞n=0Rn be a Noetherian graded ring and M = ⊕∞

n=0Mn

a finitely generated graded R-module. Then Mn is a finitely generated R0 modulefor each n ≥ 0.

Let R = ⊕∞n=0Rn be a Noetherian graded ring and M = ⊕∞

n=0Mn a finitelygenerated graded R-module.

Definition 11.3. Let λ be an additive function (namely λ(N) = λ(N ′) +λ(N ′′) forevery exact sequence 0→ N ′ → N → N ′′ → 0) on the class of finitely generated R0-modules with values in Z. The Poincaré series of M with respect to λ is P (M, t) =∑∞n=0 λ(Mn)tn ∈ Z[[t]].

It follows from additivity that λ(0) = 0.

Theorem 11.4 (Hilbert, Serre). We have P (M, t) ∈ Q(t). More precisely, if R =R0[x1, . . . , xr] with xi ∈ Rki

, then P (M, t) = f(t)/∏ri=1(1− tki) with f(t) ∈ Z[t].

We let D(M) denote the order of pole of P (M, t) at t = 1 (we put D(M) = 0 ifP (M, t) has no pole at t = 1). It is a measurement of the size of M . In the sequelwe assume ki = 1 for all i.

Definition 11.5. A numerical polynomial is a polynomial φ(z) ∈ Q[z] such thatφ(n) ∈ Z for n ∈ Z, n� 0.

Example 11.6. For d ∈ Z≥0,(zd

)= 1

d!z(z−1) · · · (z−d+1) is a numerical polynomial.Remark 11.7. One can show that every numerical polynomial φ is a Z-linear com-bination of the above. It follows that φ(n) ∈ Z for all n ∈ Z.

Corollary 11.8. Assume R = R0[R1]. Let D = D(M). Then there exists a uniquenumerical polynomial φM of degree D−1 such that φM(z) = λ(Mn) for n ≥ N+1−D,where N = deg(1− t)DP (M, t). We adopt the convention that deg(0) = −1.

17

18 CHAPTER 11. DIMENSION THEORY

Definition 11.9. The function n 7→ λ(Mn) is called the Hilbert function and φM iscalled the Hilbert polynomial.

In the sequel we assume R0 is Artinian and we take λ(N) = lg(N) to be thelength of N .Remark 11.10. In this case, for M 6= 0, P (M, t) is not zero and t = 1 is not a zeroof P (M, t). In fact, if D(M) = 0, P (M, 1) = ∑∞

n=0 lg(Mn) > 0.Example 11.11. Let R0 be an Artinian ring and let R = R0[X0, . . . , Xr], graded bydegree. Then lg(Rn) = lg(R0)

(n+rr

)′(where

(ab

)′=(ab

)for a ≥ b and

(ab

)′= 0 for

a < b.) We have φR(z) =(z+rr

)with leading term 1

r!zr.

Example 11.12. Let k be a field and F ∈ k[X0, . . . , Xr] a homogeneous polynomialof degree s. Let R = k[X0, . . . , Xr]/(F ), graded by degree. Then lg(Rn) =

(n+rr

)′−(

n−s+rr

)′, so that φR(z) =

(z+rr

)−(z−s+rr

)= ∑s

i=1

(z−i+rr−1

). The leading term is

s(r−1)!z

r−1.

Proposition 11.13. Let R = ⊕∞n=0Rn be a Noetherian graded ring with R0 Artinian

and R = R0[R1] and M 6= 0 a finitely generated graded module. Let k > 0 and x ∈Rk an M-regular element (xm = 0 implies m = 0). Then D(M/xM) = D(M)− 1.

Remark 11.14. In algebraic geometry, to a Noetherian graded ring R = ⊕∞n=0Rn

with R0 Artinian and R = R0[R1] and a finitely generated graded R-module M ,one associates a scheme Proj(R) and a coherent sheaf M on Proj(R). Then Mn 'H0(Proj(R), M(n)), where M(n)k = Mn+k, and

φM(n) =∑i

(−1)i lg(H i(Proj(R), M(n)))

is the Euler characteristic of M(n).

Dimension theory of Noetherian local ringsProposition 11.15. Let R be a Noetherian local ring of maximal ideal m, q an m-primary ideal of R, M a finitely generated R-module, F = (Mn) a stable q-filtrationof M .(1) M/Mn has finite length for all n ≥ 0.(2) There exists a unique numerical polynomial χMF such that lg(M/Mn) = χMF (n)

for n � 0. Moreover, deg(χMF ) = D(grFM) ≤ r, where r denotes the leastnumber of generators of q.

(3) The degree and leading coefficient of χMF depend only M and q, not on F .

For F = (qnM), we write χMq for χMF . For M = R, we write χq for χRq .

Corollary 11.16. There exists a unique polynomial of degree D(grqR) ≤ r suchthat lg(R/qn) = χq(n) for n� 0.

Proposition 11.17. deg(χq) = deg(χm).

19

Notation 11.18. We write d(R) for deg(χm) = D(grmR). We denote by δ(R) theleast number of generators of m-primary ideals of R.

Theorem 11.19. Let R be a Noetherian local ring. We have d(R) = dim(R) =δ(R).

We will show dim(R) ≤ d(R) and δ(R) ≤ dim(R). We start with an analogue ofProposition 11.13 for Noetherian local rings.

Proposition 11.20. Let M be a finitely generated R-module, x ∈ R such thatKer(M ×x−→M) = 0, M ′ = M/xM . Then deg(χM ′q ) ≤ deg(χMq )− 1.

Corollary 11.21. Let x ∈ R that is not a unit or zero-divisor. Then d(R/xR) ≤d(R)− 1.

We will show later that equality holds in this case (Corollary 11.33).

Proposition 11.22. dim(R) ≤ d(R).

Proposition 11.23. δ(R) ≤ dim(R).

This finishes the proof of Theorem 11.19. The dimension theorem has manyconsequences.Example 11.24. Let R0 be a nonzero Artinian ring and R = R0[X1, . . . , Xd], m =(X1, . . . , Xd). We have grmRm

(Rm) ' grm(Rm) ' R. We have seen φR(z) =lg(R0)

(z+d−1d−1

). Thus dim(Rm) = d(Rm) = D(R) = d.

Corollary 11.25. Let R be a Noetherian local ring of maximal ideal m, R its m-adiccompletion. Then dim(R) = dim(R).

Corollary 11.26. Every prime ideal in a Noetherian ring R has finite height. In ot-her words, R satisfies the descending chain condition for prime ideals. In particular,if R is a Noetherian local ring, then dim(R) <∞.

Remark 11.27. Nagata constructed a Noetherian ring R with dim(R) =∞.

Definition 11.28. The embedding dimension emb.dim(R) of a Noetherian localring R of maximal ideal m is dimk(m/m2), where k = R/m.

By Nakayama’s lemma, emb.dim(R) is the least number of generators of m.

Corollary 11.29. dim(R) ≤ emb.dim(R).

Corollary 11.30. Let R be a Noetherian ring, x1, . . . , xr ∈ R. Every isolated primeideal p belonging to (x1, . . . , xr) has height ≤ r.

The case r = 1 is called Krull’s principal ideal theorem.

Corollary 11.31. Let R be a Noetherian ring, x ∈ R not a zero-divisor. Thenevery isolated prime ideal p belonging to (x) has height 1.


Systems of parametersLet R be a Noetherian local ring of dimension d, m the maximal ideal of R.Definition 11.32. x1, . . . , xd ∈ R is called a system of parameters if (x1, . . . , xd) ism-primary.Corollary 11.33. Let x1, . . . , xr ∈ m. We have dim(R/(x1, . . . , xr)) ≥ dim(R)− r.Equality holds if x1, . . . , xr is part of a system of parameters of R.Proposition 11.34. Let x1, . . . , xd ∈ R be a system of parameters and q = (x1, . . . , xd).Let f ∈ R[X1, . . . , Xd] be a homogeneous polynomial of degree s. Assume f(x1, . . . , xd) ∈qs+1. Then f ∈ mR[X1, . . . , Xd].

Let A = R/q and α : A[X1, . . . , Xd]→ grq(R) the homomorphism carrying Xi toxi mod q. The proposition says ker(α) ⊆ mA[X1, . . . , Xd].Corollary 11.35. Assume that R has a subfield k. Then any system of parametersx1, . . . , xd is algebraically independent over k.Theorem 11.36. Let k be a field, R a finitely generated k-algebra that is a dom-ain, K = Frac(R). Then for every maximal ideal m of R, dim(R) = dim(Rm) =tr.deg(K/k), where tr.deg denotes the transcendence degree.Lemma 11.37. Let A ⊆ B be an extension of integral domains with A integrallyclosed and B integral over A. Let m be a maximal ideal of B and n = m ∩A. Thenn is maximal and dim(Bn) = dim(Am).

Regular local ringsTheorem 11.38. Let R be a Noetherian local ring of dimension d, m its maximalideal, k = R/m. The following conditions are equivalent:(1) We have an isomorphism grm(R) ' k[X1, . . . , Xd] of k-algebras.(2) dimk(m/m2) = d.(3) m is generated by d elements.

Definition 11.39. A regular local ring is a Noetherian local ring R satisfying theabove conditions. A regular system of generators for R is x1, . . . , xd such that(x1, . . . , xd) = m (where d = dim(R)).Example 11.40. (1) Regular local rings of dimension 0 are precisely fields. Regular

local rings of dimension 1 are precisely DVRs.(2) Let k be a field, R = k[X1, . . . , Xd], m = (X1, . . . , Xd). Then Rm is a regular

local ring. Indeed, grmRm(Rm) ' R.

(3) Let R be a regular local ring of dimension d and x1, . . . , xd a regular systemof parameters. Then A = R/(x1, . . . , xr) is a regular ring of dimension d− r.Indeed, xr+1, . . . , xd is a regular system of parameters for A.

Proposition 11.41. Let R be a ring, I an ideal satisfying ⋂∞n=0 In = 0. Assume

that grI(R) is a domain. Then R is a domain.Corollary 11.42. A regular local ring is a domain.Proposition 11.43. Let R be a Noetherian local ring of maximal ideal m. Then Ris regular if and only if the m-adic completion R is regular.

21

CM ringsDefinition 11.44. Let R be a ring, M an R-module. A sequence x1, . . . , xn ∈ R iscalled M-regular if it satisfies the following conditions:(1) Multiplication by xi is an injection on M/

∑i−1j=1 xjM for all 1 ≤ i ≤ n.

(2) M/∑nj=1 xjM 6= 0.

The depth of M is the supremum of the lengths of M -regular sequences.

We will only use M -regularity when R is a Noetherian local ring and M 6= 0 isa finitely generated R-module. In this case, by Nakayama’s lemma, condition (2) isequivalent to x1, . . . , xn ∈ m, where m is the maximal ideal of m.

Proposition 11.45. Let R be a Noetherian local ring, x1, . . . , xn an R-regularsequence. Then dim(R/(x1, . . . , xn)) = dim(R) − n. In particular, depth(R) ≤dim(R).

Definition 11.46. A Cohen-Macaulay (CM) local ring is a Noetherian local ringsatisfying depth(R) = dim(R).

Example 11.47. (1) Artinian local rings are CM local rings.(2) Regular local rings are CM local rings. Indeed, any regular system of parame-

ters is an R-regular sequence.Remark 11.48. One can show that if R is a regular (resp. CM) local ring, then Rp

is a regular (resp. CM) local ring for every prime ideal p.

Definition 11.49. A regular (resp. CM) ring is a Noetherian ring such that Rp isa regular (resp. CM) local ring for every prime ideal p.

Remark 11.50. A regular ring is normal. More generally, Serre proved the followingcriterion of normality: A Noetherian ring R is normal if and only if the followingconditions are satisfied:(R1) For every prime ideal p of height ≤ 1, Rp is regular.(S2) For every prime ideal p of height ≥ 2, depth(Rp) ≥ 2.

Summary of properties of rings

field +3

��

DVR or field +3

��

valuation ring

%-

+3 local ring

PID or field +3

��

UFD +3 normal ring

Dedekind domain or field +3 regular ring

19

��Artinian ring +3 CM ring +3 Noetherian ring

23

24 SUMMARY OF PROPERTIES OF RINGS

Bibliography

[AM] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-WesleyPublishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR0242802 ↑3

[B1] N. Bourbaki, Éléments de mathématique. Topologie générale, Springer-Verlag, 2007 (French).↑12

[B2] , Éléments de mathématique. Algèbre commutative, Springer-Verlag, 2006 (French).↑8

[E] D. Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR1322960 ↑

[G] A. Grothendieck, Éléments de géométrie algébrique (avec la collaboration de J. Dieudonné).IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ.Math. 20, 24, 28, 32 (1964) (French). MR0173675, MR0199181, MR0217086, MR0238860↑7

[L] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189,Springer-Verlag, New York, 1999. MR1653294 (99i:16001) ↑

[M1] H. Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56,Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR575344 ↑7

[M2] , Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8,Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid.MR1011461 ↑7

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